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7.5.6 The Homotopy Colimit as a Derived Functor

Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. In ยง7.5.3, we showed that the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the limit of an isofibrant replacement for $\mathscr {F}$: that is, there exists an isomorphism $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \varprojlim ( \mathscr {F}^{+} )$, where $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram equipped with a levelwise categorical equivalence $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ (Construction 7.5.3.3 and Proposition 7.5.3.7). Our goal in this section is to present a parallel treatment of the homotopy colimit functor of Construction 5.3.2.1. More precisely, we show that the homotopy colimit of a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ can be identified with the colimit of an auxiliary diagram $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ which is equipped with a levelwise weak homotopy equivalence $\mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ (Proposition 7.5.6.12).

We begin by introducing some terminology. Recall that a natural transformation $\beta : \widetilde{\mathscr {G}} \rightarrow \mathscr {G}$ is a levelwise trivial Kan fibration if, for each object $C \in \operatorname{\mathcal{C}}$, the morphism $\beta _{C}: \widetilde{\mathscr {G}}(C) \rightarrow \mathscr {G}(C)$ is a trivial Kan fibration of simplicial sets.

Definition 7.5.6.1. Let $\operatorname{\mathcal{C}}$ be a small category. We say that a diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if, for every levelwise trivial Kan fibration $\beta : \mathscr {G}' \rightarrow \mathscr {G}$, the induced map

\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}) \]

is surjective. That is, every natural transformation $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ factors through $\beta $.

Example 7.5.6.2. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then the diagram

\[ \mathscr {F}_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {F}_{\operatorname{\mathcal{E}}}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]

is projectively cofibrant, in the sense of Definition 7.5.6.1. To prove this, we must show that for every levelwise trivial Kan fibration $\mathscr {G}' \rightarrow \mathscr {G}$ between functors $\mathscr {G}', \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the induced map

\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}_{\operatorname{\mathcal{E}}}, \mathscr {G}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {G}_{\operatorname{\mathcal{E}}}, \mathscr {G} ) \]

is surjective. Using Proposition 5.3.3.21, we can identify $\theta $ with a pullback of the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G}'}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G} }(\operatorname{\mathcal{C}}) )$, which is surjective by virtue of Exercise 5.3.3.11.

Exercise 7.5.6.3 (Well-Founded Diagrams). Let $(Q, \leq )$ be a well-founded partially ordered set. Show that a diagram of simplicial sets $\mathscr {F}: Q \rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if and only if, for each element $q \in Q$, the associated map $\varinjlim _{ p < q} \mathscr {F}(p) \rightarrow \mathscr {F}(q)$ is a monomorphism of simplicial sets (compare with Proposition 4.5.6.6).

Example 7.5.6.4 (Projectively Cofibrant Sequences). A sequential diagram of simplicial sets

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow X(3) \rightarrow \cdots \]

is projectively cofibrant (when regarded as a functor $\operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{Set_{\Delta }})$ if and only if each of the transition maps $X(n) \rightarrow X(n+1)$ is a monomorphism.

Example 7.5.6.5 (Projectively Cofibrant Squares). A commutative diagram of simplicial sets

7.52
\begin{equation} \begin{gathered}\label{equation:projectively-cofibrant-squares} \xymatrix { A \ar [r]^{f_0} \ar [d]^{f_1} & A_0 \ar [d]^{f_1} \\ A_1 \ar [r]^{f_0} & A_{01} } \end{gathered} \end{equation}

is projectively cofibrant (when regarded as a functor $[1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$) if and only if the morphisms

\[ f_0: A \rightarrow A_0 \quad \quad f_1: A \rightarrow A_1 \quad \quad (f'_1, f'_0): A_0 \coprod _{A} A_1 \rightarrow A_{01} \]

are monomorphisms of simplicial sets. Equivalently, (7.52) is projectively cofibrant if it is a pullback square consisting of monomorphisms.

Remark 7.5.6.6 (Relationship to Isofibrant Diagrams). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\operatorname{\mathcal{D}}$ be an $\infty $-category, and let $\operatorname{\mathcal{D}}^{\mathscr {F}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by the construction $C \mapsto \operatorname{Fun}( \mathscr {F}(C), \operatorname{\mathcal{D}})$. If $\mathscr {F}$ is projectively cofibrant (in the sense of Definition 7.5.6.1), then $\operatorname{\mathcal{D}}^{\mathscr {F}}$ is isofibrant (in the sense of Definition 4.5.6.3). That is, if $\mathscr {E}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ is a diagram of simplicial sets and $\mathscr {E}_0 \subseteq \mathscr {E}$ is a subfunctor for which the equivalence $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence, then the restriction map

\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \operatorname{\mathcal{D}}^{\mathscr {F}} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \operatorname{\mathcal{D}}^{\mathscr {F}} ) \]

is surjective. This follows from the observation that $\theta $ can be identified with the map

\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{\mathcal{D}}^{\mathscr {E}} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{\mathcal{D}}^{\mathscr {E}_0} ) \]

given by composition with the restriction map $\operatorname{\mathcal{D}}^{\mathscr {E}} \rightarrow \operatorname{\mathcal{D}}^{\mathscr {E}_0}$, which is a levelwise trivial Kan fibration by virtue of Corollary 4.5.5.19.

Proposition 7.5.6.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between projectively cofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\alpha $ is a levelwise categorical equivalence, then the induced map $\varinjlim (\alpha ): \varinjlim ( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {G} )$ is a categorical equivalence of simplicial sets. If $\alpha $ is a levelwise weak homotopy equivalence, then $\varinjlim (\alpha )$ is a weak homotopy equivalence.

Proof. We will prove the first assertion; the second follows by a similar argument. Assume that $\alpha $ is levelwise categorical equivalence and let $\operatorname{\mathcal{D}}$ be an $\infty $-category; we wish to show that precomposition with $\varprojlim (\alpha )$ induces an equivalence of $\infty $-categories $\alpha ^{\ast }: \operatorname{Fun}( \varinjlim ( \mathscr {G} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \varinjlim ( \mathscr {F}), \operatorname{\mathcal{D}})$. $\alpha $ is a levelwise categorical equivalence, precomposition with $\alpha $ induces a levelwise categorical equivalence $\beta : \operatorname{\mathcal{D}}^{ \mathscr {G} } \rightarrow \operatorname{\mathcal{D}}^{\mathscr {F}}$ in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }})$. Unwinding the definitions, we see that $\alpha ^{\ast }$ can be identified with the limit $\varprojlim (\beta )$. Since $\operatorname{\mathcal{D}}^{\mathscr {F}}$ and $\operatorname{\mathcal{D}}^{\mathscr {G}}$ are isofibrant diagrams (Remark 7.5.6.6), the functor $\varprojlim (\beta )$ is an equivalence of $\infty $-categories (Corollary 4.5.6.15). $\square$

We now show that every diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ admits a weak homotopy equivalence from a projectively cofibrant diagram (for a stronger statement, see Proposition 7.5.9.7).

Construction 7.5.6.8 (Explicit Cofibrant Replacement). Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ denote the homotopy colimit of $\mathscr {F}$ (Construction 5.3.2.1). For each object $C \in \operatorname{\mathcal{C}}$, we let $\mathscr {F}_{+}(C)$ denote the simplicial set given by the fiber product

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ /C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ). \]

The construction $C \mapsto \mathscr {F}_{+}(C)$ determines a diagram of simplicial sets $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. This diagram is equipped with a natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the comparison map

\[ \mathscr {F}_{+}(C) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \twoheadrightarrow \varinjlim ( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \simeq \mathscr {F}(C) \]

of Remark 5.3.2.9.

Proposition 7.5.6.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the diagram $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 7.5.6.8 is projectively cofibrant, and the natural transformation $\alpha : \mathscr {F}_{+} \rightarrow \mathscr {F}$ is a levelwise weak homotopy equivalence. Moreover, $\alpha $ is also an epimorphism.

Proof. Example 7.5.6.2 shows that the diagram $\mathscr {F}_{+}$ is projectively cofibrant and Remark 5.3.2.9 shows that $\alpha $ is an epimorphism. To complete the proof, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the map $\alpha _{C}: \mathscr {F}_{+}(C) \rightarrow \mathscr {F}(C)$ is a weak homotopy equivalence of simplicial sets. Replacing $\operatorname{\mathcal{C}}$ by the slice category $\operatorname{\mathcal{C}}_{/C}$, we can reduce to the case where $C$ is a final object of $\operatorname{\mathcal{C}}$; in this case, we wish to prove that the comparison map

\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {F} ) \simeq \mathscr {F}(C) \]

is a weak homotopy equivalence. Note that this map admits a section, given by the inclusion map

\[ \iota : \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]

We complete the proof by that our assumption that $C \in \operatorname{\mathcal{C}}$ is a final object guarantees that $\iota $ is right anodyne (Example 7.2.3.12). $\square$

Warning 7.5.6.10. In the situation of Proposition 7.5.6.9, the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ is usually not a levelwise categorical equivalence. For example, if $\mathscr {F}$ is the constant functor taking the value $\Delta ^0$, then $\mathscr {F}_{+}$ is given by the construction $C \mapsto \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$.

Remark 7.5.6.11. Constructions 7.5.6.8 and 7.5.3.3 are closely related. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Combining Corollary 5.3.2.24 with Proposition 5.3.3.21, we obtain canonical isomorphisms of Kan complexes

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}^{+} )_{\bullet } & = & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})/\operatorname{\mathcal{C}}} ) \\ & \simeq & \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}_{+}, \mathscr {G} )_{\bullet }. \end{eqnarray*}

More generally, if $\mathscr {G}$ is a diagram of $\infty $-categories, we can identify $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}^{+} )_{\bullet }$ with the full subcategory of $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}_{+}, \mathscr {G} )_{\bullet }$ spanned by those natural transformations $\alpha : \mathscr {F}_{+} \rightarrow \mathscr {G}$ having the property that, for each object $C \in \operatorname{\mathcal{C}}$, the diagram

\[ \alpha _{C}: \mathscr {F}_{+}(C) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \rightarrow \mathscr {G}(C) \]

carries horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } )$ to isomorphisms in the $\infty $-category $\mathscr {G}(C)$.

Proposition 7.5.6.12. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the diagram of Construction 7.5.6.8. Then there is a canonical isomorphism of simplicial sets $\lambda : \varinjlim ( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ which is characterized by the following requirement: for each object $C \in \operatorname{\mathcal{C}}$, the composition

\begin{eqnarray*} \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) & = & \mathscr {F}_{+}(C) \\ & \rightarrow & \varinjlim ( \mathscr {F}_{+} ) \\ & \xrightarrow {\lambda } & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \end{eqnarray*}

is given by projection onto the second factor.

Proof. It follows from the definition of the colimit that there is a unique morphism of simplicial sets $\lambda : \varinjlim ( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ having the desired property. Using the dual of Lemma 7.5.3.8, we deduce that $\lambda $ is an isomorphism. $\square$

Remark 7.5.6.13. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\theta : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ be the comparison map of Remark 5.3.2.9, and let $\lambda : \varinjlim ( \mathscr {F}_{+} ) \xrightarrow {\sim } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ be the isomorphism of Proposition 7.5.6.12. Then the composition $(\theta \circ \lambda ): \varinjlim ( \mathscr {F}_{+} ) \rightarrow \varinjlim ( \mathscr {F} )$ is induced by the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ appearing in Construction 7.5.6.8.

Corollary 7.5.6.14. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a projectively cofibrant diagram of simplicial sets. Then the comparison map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ of Remark 5.3.2.9 is a weak homotopy equivalence.

Proof. By virtue of Remark 7.5.6.13, it will suffice to show that the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ of Construction 7.5.6.8 induces a weak homotopy equivalence $\varinjlim (\alpha ): \varinjlim ( \mathscr {F}_{+} ) \rightarrow \varinjlim ( \mathscr {F} )$. This is a special case of Proposition 7.5.6.7, since $\alpha $ is a levelwise weak homotopy equivalence between projectively cofibrant diagrams (Proposition 7.5.6.9). $\square$

Warning 7.5.6.15. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of simplicial sets, let $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ be the natural transformation of Construction 7.5.6.8, and let $\lambda : \varinjlim ( \mathscr {F}_{+} ) \xrightarrow {\sim } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ be the isomorphism of Proposition 7.5.6.12. Then we have a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) \ar [r]^-{ \underset { \longrightarrow }{\mathrm{holim}}( \alpha ) } \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \varinjlim ( \mathscr {F}_{+} ) \ar [r]_{\varinjlim (\alpha ) } \ar [ur]^{\lambda }_{\sim } & \varinjlim ( \mathscr {F} ), } \]

where the outer square and the lower right triangle are commutative (Remark 7.5.6.13). Beware that the upper left triangle is usually not commutative. That is, $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varinjlim ( \mathscr {F}_{+} )$ are isomorphic when viewed as abstract simplicial sets, but not when viewed as quotients of the simplicial set $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} )$ (compare with Warning 7.5.3.14).

Remark 7.5.6.16 (The Homotopy Colimit as a Left Derived Functor). The preceding results can be interpreted in the language of model categories. For every small category $\operatorname{\mathcal{C}}$, the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ can be equipped with a model structure in which the fibrations are levelwise Kan fibrations and weak equivalences are levelwise weak homotopy equivalences (see Example ). Combining Propositions 7.5.6.9 and 7.5.6.12, we deduce that the homotopy colimit functor $ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$ can be viewed as a left derived functor of the usual colimit $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$ (see Definition ).